Oswalt has thrown for four teams in the past two weeks, Rosenthal reports, signaling how much he wants to return to the majors. But though teams are intrigued, they remain turned off by his asking price. One executive Rosenthal spoke to said that price is "always high." Another source said it was $7.5 million, which turned the Dodgers away. Oswalt had hoped to begin pitching this season by June 15.

Oswalt declined the Tigers' $10 million offer during his first round of free agency, and he said he would wait until the season to sign with a team. Considering his original target of June 15, though, he will likely push to sign soon.

Oswalt started 23 games last season for the Phillies, going 9-10 with a 3.69 ERA. But he was hampered during the season by back problems, twice having to go on the disabled list.